For deer hunting, you want to go West. Almost all hunting land are private leases that you have to pay to hunt on. Going out to the Del Rio are, or even around Sanderson has great hunting. Around Sanderson, TX is popular due to the mix of both mule and white tail deer and the fact that leases are pretty inexpensive due to the fact its in the middle of nowhere.
For birding, I prefer the Gulf Coast region, down near Edna, or El Campo. We go hunting on family land for dove and turkey, but there are a ton of leases.
Don't forget to go to Academy and get your license.
Maybe this could help: they might list public land to hunt on:
http://www.tpwd.state.tx.us/huntwild/
I recommend the Super Combo license that gets you your fishing license as well. That way you can go to the Matagorda area and fish the Redfish, Trout, and Flounder as well. I recommend Port O'Conner for good guides for bay and deep sea fishing as well.